Stroke patients with ideomotor apraxia (IMA) have difficulties controlling voluntary motor actions, as clearly seen when asked to imitate simple gestures performed by the examiner. Despite extensive research, the neurophysiological mechanisms underlying failure to imitate gestures in IMA remain controversial. The aim of the current study was to explore the relationship between imitation failure in IMA and mirror neuron system (MNS) functioning. Mirror neurons were found to play a crucial role in movement imitation and in imitation-based motor learning. Their recruitment during movement observation and execution is signaled in EEG recordings by suppression of the lower (8–10 Hz) mu range. We examined the modulation of EEG in this range in stroke patients with left (n = 21) and right (n = 15) hemisphere damage during observation of video clips showing different manual movements. IMA severity was assessed by the DeRenzi standardized diagnostic test. Results showed that failure to imitate observed manual movements correlated with diminished mu suppression in patients with damage to the right inferior parietal lobule and in patients with damage to the right inferior frontal gyrus pars opercularis—areas where major components of the human MNS are assumed to reside. Voxel-based lesion symptom mapping revealed a significant impact on imitation capacity for the left inferior and superior parietal lobules and the left post central gyrus. Both left and right hemisphere damages were associated with imitation failure typical of IMA, yet a clear demonstration of relationship to the MNS was obtained only in the right hemisphere damage group. Suppression of the 8–10 Hz range was stronger in central compared with occipital sites, pointing to a dominant implication of mu rather than alpha rhythms. However, the suppression correlated with De Renzi's apraxia test scores not only in central but also in occipital sites, suggesting a multifactorial mechanism for IMA, with a possible impact for deranged visual attention (alpha suppression) beyond the effect of MNS damage (mu suppression).